Trigonometry Examples

$\sec (2 x) = \frac{\csc^{2} (x)}{2 \cot^{2} (x) - \csc^{2} (x)}$

Step 1

Start on the right side.

$\frac{\csc^{2} (x)}{2 \cot^{2} (x) - \csc^{2} (x)}$

Step 2

Simplify the expression.

Tap for more steps...

Step 2.1

Simplify the numerator.

Tap for more steps...

Step 2.1.1

Rewrite $\csc (x)$ in terms of sines and cosines.

$\frac{{(\frac{1}{\sin (x)})}^{2}}{2 \cot^{2} (x) - \csc^{2} (x)}$

Step 2.1.2

Apply the product rule to $\frac{1}{\sin (x)}$ .

$\frac{\frac{1^{2}}{\sin^{2} (x)}}{2 \cot^{2} (x) - \csc^{2} (x)}$

Step 2.1.3

One to any power is one.

$\frac{\frac{1}{\sin^{2} (x)}}{2 \cot^{2} (x) - \csc^{2} (x)}$

Step 2.2

Simplify the denominator.

Tap for more steps...

Step 2.2.1

Rewrite $\cot (x)$ in terms of sines and cosines.

$\frac{\frac{1}{\sin^{2} (x)}}{2 {(\frac{\cos (x)}{\sin (x)})}^{2} - \csc^{2} (x)}$

Step 2.2.2

Apply the product rule to $\frac{\cos (x)}{\sin (x)}$ .

$\frac{\frac{1}{\sin^{2} (x)}}{2 \frac{\cos^{2} (x)}{\sin^{2} (x)} - \csc^{2} (x)}$

Step 2.2.3

Combine $2$ and $\frac{\cos^{2} (x)}{\sin^{2} (x)}$ .

$\frac{\frac{1}{\sin^{2} (x)}}{\frac{2 \cos^{2} (x)}{\sin^{2} (x)} - \csc^{2} (x)}$

Step 2.2.4

Rewrite $\csc (x)$ in terms of sines and cosines.

$\frac{\frac{1}{\sin^{2} (x)}}{\frac{2 \cos^{2} (x)}{\sin^{2} (x)} - {(\frac{1}{\sin (x)})}^{2}}$

Step 2.2.5

Apply the product rule to $\frac{1}{\sin (x)}$ .

$\frac{\frac{1}{\sin^{2} (x)}}{\frac{2 \cos^{2} (x)}{\sin^{2} (x)} - \frac{1^{2}}{\sin^{2} (x)}}$

Step 2.2.6

One to any power is one.

$\frac{\frac{1}{\sin^{2} (x)}}{\frac{2 \cos^{2} (x)}{\sin^{2} (x)} - \frac{1}{\sin^{2} (x)}}$

Step 2.2.7

Combine the numerators over the common denominator.

$\frac{\frac{1}{\sin^{2} (x)}}{\frac{2 \cos^{2} (x) - 1}{\sin^{2} (x)}}$

Step 2.2.8

Apply the cosine double-angle identity.

$\frac{\frac{1}{\sin^{2} (x)}}{\frac{\cos (2 x)}{\sin^{2} (x)}}$

Step 2.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{\sin^{2} (x)} \cdot \frac{\sin^{2} (x)}{\cos (2 x)}$

Step 2.4

Combine.

$\frac{1 \sin^{2} (x)}{\sin^{2} (x) \cos (2 x)}$

Step 2.5

Cancel the common factor of $\sin^{2} (x)$ .

Tap for more steps...

Step 2.5.1

Cancel the common factor.

$\frac{1 \sin^{2} (x)}{\sin^{2} (x) \cos (2 x)}$

Step 2.5.2

Rewrite the expression.

$\frac{1}{\cos (2 x)}$

Step 3

Rewrite $\frac{1}{\cos (2 x)}$ as $\sec (2 x)$ .

$\sec (2 x)$

Step 4

Because the two sides have been shown to be equivalent, the equation is an identity.

$\sec (2 x) = \frac{\csc^{2} (x)}{2 \cot^{2} (x) - \csc^{2} (x)}$ is an identity